arXiv:1311.5862v1 [math.DG] 22 Nov 2013
A Survey of the Differential Geometry of
Discrete Curves
Daniel Carroll, Eleanor Hankins, Emek Kose, Ivan Sterling
November 25, 2013
1
Introduction
Discretization of curves is an ancient topic. Even discretization of curves with
an eye toward differential geometry is over a century old. However there is no
general theory or methodology in the literature, despite the ubiquitous use of
discrete curves in mathematics and science. There are conflicting definitions
of even basic concepts such as discrete curvature κ, discrete torsion τ , or
discrete Frenet frame.
Consider for example the three equally worthy definitions of the curvature
of an angle derived in Section 2 by considering the problem of approximating
an N-gon (by N-gon we mean a regular N-gon) with sides of length ` by a
circle:
2
θ
2
θ
θ
κ = sin , κ = tan , κ = .
(1)
`
2
`
2
`
In the literature each of these definitions occur frequently. For example [9],
[7], [6]. As we show, the source of this variety is that each author chooses
whether to normalize their curvature by using the inscribed, circumscribed
or centered circle of an N-gon. Although our initial interest was in particular
applications, we realized the need for a general approach and along the way
discovered some pleasing theorems.
Using Section 2 as a guide we proceed to build three theories of discrete
1
curves all of which culminate in a discrete version of the Frenet equations:
DT e =
κN v ,
DN e = −κT v
+ τ Bv ,
DB e =
−τ N v .
(2)
Although dozens of discrete Frenet equations can be found in the literature,
all have unpleasant error terms. Our approach is new, and the resulting
Equations (2) are free any error terms. We also show that our definitions of
discrete length `, curvature κ and τ reproduce a unique (up to rigid motion)
discrete curve with the given `, κ and τ .
In each of the three cases – inscribed, circumscribed, and centered – there
corresponds a natural differential geometric way to define the discretization
of a smooth curve. These definitions are discussed in Section 4. Conversely
given a discrete curve there is a natural differential geometric way to spline
the curve. See Section 5. Of particular interest is that discrete curves in the
plane R2 are naturally splined by special piecewise curves: constant curvature
in the inscribed case, clothoids in the circumscribed case, and elastic curves
in the centered case. In each case we argue for our definition by showing
that these splines are the constrained minimizers of the natural variable in
that case. Section 6 contains some brief comments about applications and
discrete surface theory.
2
Circles and N-gons
For every N ≥ 3, a circle has an inscribed, circumscribed and centered Ngon which discretizes it. An N-gon is called centered about a circle if its
perimeter equals the circumference of the circle. Conversely an N-gon has an
inscribed, a circumscribed and a centered circle which splines it. See Figure
1 for the case N = 4. The nomenclature can be confusing, so care must be
taken. For example: “An N-gon is centered about a circle” means the circle
is given first, whereas “a circle is inscribed inside an N-gon” means the N-gon
is given first. From trigonometry, Figure 2, we are led to three definitions
of curvature for a given N-gon. Recall the curvature of a circle is defined by
. We assume all
κ = 1/r and that the exterior angle for an N-gon is θ = 2π
N
sides have length ` and N is any real N > 0. We have the curvature of the
2
Figure 1: Discretizing a Circle, Splining an N-gon
circle inscribed in an N-gon is
κ=
2
θ
sin .
`
2
(3)
Similarly the curvature of the circle circumscribing an N-gon is
κ=
θ
2
tan ,
`
2
(4)
and in the centered case we have
θ
κ= .
`
(5)
We will use these basic formulas to guide us in all the definitions that
follow. In a way that will be made more precise below we consider θ as the
measure of the angle between neighboring “tangent vectors”. θ measures
the turning of an N-gon at a vertex. For a discrete curve in three space if
we similarly define φ to measure the angle between neighboring “binormal
vectors” then φ measures the twisting of a discrete curve along its edge.
See Figures 3 and 5. We define the curvature at a vertex of a discrete curve
3
{
{
{
Θ
r
Θ
Θ
r
Ȑ2
Ȑ2
r
Figure 2: Trigonometry for curvature of an N-Gon
Bp
Bp
Bq
Tq
Φ
Bq
p
Θ
p
Tp
q
q
Tp
Figure 3: θ Measures Turning, φ Measures Twisting
4
in three space by Equations (3), (4), and (5). We are similarly led to define
the torsion at a vertex by
 2
φ
 ` sin 2 , in the inscribed case,
2
τ :=
tan φ2 , in the circumscribing case,
 φ`
, in the centered case.
`
3
Discrete Frenet Equations
A discrete map is a function with domain Z, χ : Z −→ R. Such a map is
called a discrete function (resp. curve) if the range is R (resp. R3 ). Since
we work exclusively with these special ranges, we will use without further
comment the standard operations of R and R3 . If χ : Z −→ R, then we
often use the notation χi := χ(i). We define discrete differentiation (resp.
addition) by (Dχ)i := χi+1 − χi (resp. (M χ)i := χi+1 + χi ).
3.1
Frenet Frames
We will define the lengths `i , curvatures κi and torsions τi of discrete curves
in such a way that given any `i , κi , τi it is possible to reconstruct a discrete
curve with these lengths, curvatures and torsions. We will also require that a
natural discrete version of the Frenet equations hold. As we have seen, there
are at least three reasonable definitions of the curvature of the elementary
N-gon. We will investigate these three cases using the definitions of curvature
and torsion derived from the formulas above.
Let γ orig be a discrete curve
γ orig : Z −→ R3 ,
which we call “the original curve.” Then we define the curve γ : Z −→ R3 as
follows. See Figure 4 where the larger numbers are the indices for the original
curve and the smaller numbers are the indices for the redefined curve. First
we define
i−1
orig
if i is odd
γ(i) := γ
2
and then
γ(i) :=
γ(i + 1) + γ(i − 1)
if i is even.
2
5
3
7
1
3
4
2
6
5
2
1
0
-3
0
-2
-2
-1
-1
Figure 4: Original and Redefined Discrete Curve
Note that we recover the original curve from the odd indices of γ and that the
even indices are mapped to the midpoints of the original curve. We define
the discrete length by
`i := k(Dγ)i k.
γ is parametrized by arc length if ` ≡ 1 and it is parametrized proportional
to arc length if ` is constant. Note that `γ ≡ ` = constant if `γ orig ≡ 2`. For
clarity of presentation we will assume from now on that γ is parametrized
proportional to arc length, kDγk ≡ ` = constant. The theory goes through
without this restriction.
3.2
Frenet Equations
In each version (Inscribed, Circumscribed and Centered) we will produce two
discrete Frenet frames {T e , N e , B e } and {T v , N v , B v }. First for {T e , N e , B e }:
T e :=
Dγ
Dγ
=
.
kDγk
`
6
e
Note Tie = Ti−1
if i is even. Then
Bie :=
e
Tie × Ti+1
if i is even
e
kTie × Ti+1
k
and
e
Bie := Bi−1
if i is odd.
and finally for all i:
Nie := Bie × Tie .
For {T v , N v , B v } we have for all i:
Tiv
(M T e )i
,
:=
k(M T e )i k
Biv :=
(M B e )i
,
k(M B e )i k
Niv :=
(M N e )i
.
k(M N e )i k
Note for all i, Niv = Biv × Tiv .
As shown again in Figure 5 the frame is turning, about the axis determined by the binormal, at the “vertices”. The frame is twisting, about the
axis determined by the tangent, at the “edges”. It was precisely this alternating approach which lead to the elegant form of the discrete Frenet equations
(6) given below; which do not appear in the literature.
7
B0 e
B0 e
B2 e
N2 e
Φ
N1 e
N1 e
N0 e
T1 e
Θ
T1 e
T0 e
Figure 5: Bird’s Eye View
3.3
Curvature and Torsion
The positively oriented frames {Tie , Nie , Bie } determine orientations of {Tie , Nie }
e
and note that
and {Nie , Bie }. We define θi as the angle between Tie and Ti+1
e
e
θi = 0 if i is odd. We define φi as the angle between Bi and Bi+1 with φi = 0
if i is even. To avoid technical details we will assume θi , φi ∈ [0, π2 ].
The curvature κ is defined by


kDT e k = 2` sin 2θ , in the inscribed case,





kDT e k
= 2` tan 2θ , in the circumscribing case,
κ :=
kM T e k





 2 sin−1 ( kDT e k ) = θ , in the centered case.
2
`
Note that κi = 0 if i is odd. Similarly the torsion τ is defined by


φ kDB e k = 2` sin φ2 , in the inscribed case,





kDB e k
= 2` tan φ2 , in the circumscribing case,
τ :=
kM B e k





 2 sin−1 kDB e k = φ , in the centered case.
2
`
With τi = 0 if i is even.
8
3.4
Discrete Frenet Equations
In each version (Inscribed, Circumscribed and Centered) a direct calculation
shows that the discrete Frenet equations hold.
Theorem 3.1
DT e =
κN v ,
DN e = −κT v
+ τ Bv ,
DB e =
−τ N v .
3.5
(6)
Discrete Fundamental Theorem
On the other hand we can reconstruct the curve by the relations:
e
Ti+1
= cos θi Tie +
sin θi Nie ,
e
Ni+1
= − sin θi Tie +sin(θi + φi )Nie − sin φi Bie ,
e
Bi+1
=
cos φi Nie + sin φi Bie .
e
.
and γi+1 = γi + Ti+1
To summarize we have
Theorem 3.2 Given θi , φi with θi = 0 for i odd and φi = 0 for i even.
Then for arbitrary initial conditions γ0 , T0e , N0e , B0e there exists a unique discrete curve γ with θiγ = θi , φγi = φi satisfying γ(0) = γ0 , T γ e0 = T0e , N γ e0 =
N0e , B γ e0 = B0e . Moreover, γ orig (i) := γ(2i) satisfies kDγ orig k = 2`.
4
2D-Discretizing
Given a curve in R2 we would now like to discretize it. There is a canonical
geometric discretization in each of our three cases.
4.1
Inscribed 2D-Discretization
The only distinguishing feature in this case is that each vertex of the discretization be on the curve itself. Thus any increasing map ι : Z −→ R will
produce an acceptable discrete curve δ := γ ◦ ι. See Figure 6.
9
1
-6
-4
-2
2
4
6
-1
-2
-3
-4
Figure 6: Inscribed Discretization
4.2
Circumscribed 2D-Discretization
If there are no inflection points then we again take any increasing map ι :
Z −→ R such that consecutive tangents are not parallel. We require that the
edges of our discrete curve δ intersect tangentially with the given curve at
the points (γ ◦ ι)i . We define δi to be the unique intersect point of tangent
lines at (γ ◦ ι)i and (γ ◦ ι)i+1 as in Figure 7.
2
1
-6
-4
-2
2
4
6
-1
-2
-3
-4
-5
Figure 7: Circumscribed Discretization
If there are isolated inflection points then they, as well as at least one point
in-between them, need to be included in the set of tangent points. If a curve
has infinitely many inflection points on a finite interval, then our algorithm
fails.
10
4.3
Centered 2D-Discretization
The natural centered discretization of a curve requires a bit more finesse.
First, without loss of generality, we assume γ is parametrized by arc length
and require our discretization to be parametrized proportional to arc length.
Secondly, with loss of generality, we assume γ has no inflection points, say
κ > 0 everywhere. We require κi > 0 for our discretization. Finally we choose
M “large enough.” We take the specific ι : Z −→ R defined by ι(i) := Mi and
let δ start = γ ◦ ι. For each i we offset δistart along the (outward) normal to γ
at δistart by the amount
ki
M
ki
− sin M
offseti :=
,
ki
ki sin M
where ki is the curvature of γ at δistart and we assume ki > 0. Note that this
formula is derived from the case of centered N-gons discretizing a circle.
2
-6
-4
-2
2
4
6
-2
-4
-6
Figure 8: Offset Discretization
To see the offset more clearly we zoom into the center of the curve.
11
1.0
0.8
0.6
0.4
0.2
- 1.0
- 0.5
0.5
1.0
- 0.2
Figure 9: Offset Discretization Zoom
These offset points will be the even vertices, δ2j , of our final discrete curve
δ. Now we consider the condition that our discrete curve δ is to have the
same length as our original curve γ. With the additional conditions that
i
, kδ2j+2 − δ2j+1 k and κ2j+1 > 0; we see there is one and
kδ2j+1 − δ2j k = 2M
only one way to achieve this. See Figure 10.
2
-6
-4
-2
2
4
-2
-4
-6
Figure 10: Centered Discretization
Again, we see more detail by zooming in, Figure 11.
12
6
1.0
0.8
0.6
0.4
0.2
- 1.0
- 0.5
0.5
1.0
- 0.2
Figure 11: Centered Discretization Zoom
To include inflection points requires more general parametrizations and we
will leave it as an exercise for the reader.
5
Geometric Splinings of Discrete Curves
2.0
1.5
1.0
0.5
0.5
1.0
1.5
2.0
2.5
Figure 12: Discrete Curve to be Splined
5.1
What does Best Spline Mean?
In non-geometric splining, the best spline is usually related to the degree
of the polynomial γ(t) = (x(t), y(t)) used to approximate the curve. For
example a cubic spline is constructed using piece-wise cubic polynomials.
Typically a cubic spline passes through the points of a discrete curve with
certain boundary conditions. Geometric splinings on the other hand are
13
found by considering curves whose curvature function κ(t) is
R a2 low degree
polynomial. Alternatively a best geometric spline minimizes κ .
5.2
Inscribed Splining
An inscribing spline is one which tangentially goes through the midpoints
of the edges of the given discrete curve. We seek a curve whose curvature
has the lowest degree possible. Because we are assuming our discrete curves
are parametized proportional to arc length there is a trivial differentiable
inscribed splining by pieces of curves of constant curvature. That is pieces of
circles. See Figure 13. If our discrete curve is not parametrized proportional
to arc length, then the inscribed splining would require clothoids, which are
described in the next subsection.
2.5
2.0
1.5
1.0
0.5
-0.5
0.5
1.0
1.5
2.0
2.5
3.0
Figure 13: Inscribed Splining
Notice the curvature jumps at the midpoints so our splining is not twice
differentiable.
5.3
Clothoids
Curves with linear curvature are called first order clothoids. Given
κ(s) = as + b
(if a = 0, we get a piece of a circle, a “zeroth order clothoid”) then the
turning angle θ is given by
Z s
θ(s) =
κ(t) dt + θ0 .
0
14
First order clothoids are given in terms of Fresnel integrals
Z s
Z s
sin θ(t) dt + y0 .
γ(s) =
cos θ(t) dt + x0 ,
0
0
Similarly curves with quadratic curvature are second order clothoids, and so
on.
5.4
Circumscribed Splining
A circumscribing spline is one which differentiable goes through the points of
the given discrete curve. Unlike the case of inscribed splinings, it will rarely
be the case that a circumscribed splining will consist of piece of circles. On
the other hand there will always be circumscribing splining, as in Figure 14,
using first order clothoids. If there is more than one, we take the shortest
one. This is called the fitting problem. See for example [3].
2.5
2.0
1.5
1.0
0.5
-0.5
0.5
1.0
1.5
2.0
2.5
3.0
Figure 14: Circumscribed Splining
Notice again the curvature jumps at the midpoints.
5.5
Centered Splining
For the centered spline we first offset the vertices using the centered circles
of N-gons and take the directions of the desired spline at these offset point
to be the average of the incoming and outgoing directions of the edges at the
vertices. See Figure 15.
15
2.0
1.5
1.0
0.5
0.5
1.0
1.5
2.0
2.5
Figure 15: Centered Splining Offsets
We then seek differentiable splines passing through these offset points
whose lengthR agrees with that of the given discrete curve. These curves are
found using κ2 and are called elastica. These are solutions to a variational
problem proposed by Bernoulli to Euler in 1744; that of minimizing the
bending energy of a thin inextensible wire. Among all curves of the same
length that not only pass through points A and B but are also tangent to
given straight lines at these
R 2 points, it is defined as the one minimizing the
value of the expression κ .
The one parameter family of elastic curves introduced by Euler [10] is well
known. They are all given by explicit formulas involving elliptic integrals.
These formulas arise by solving the one-dimensional sine-Gordon differential
equation. Which is alternatively written as θ00 = sin θ or [8] θ000 + 21 (θ0 )3 +Cθ0 =
0. In applied problems, such as finding the elastic curve with the boundary
conditions care must be taken for several reasons. One issue is that there
are several types of “elastic intervals” (inflectional, non-inflectional, critical,
circular, and linear). Another issue is that in some cases there are multiple
solutions. An excellent survey of the subject is in Andentov [1]. As discussed
in [1] these problems persist when attempting to numerically approximate
elastic curves.
Sogo [11] shows how, at least in some cases, “integrable discretization”
theory can be used to construct a discretized one-dimensional sine-Gordon
equation satisfied by discretized elliptic integrals. For example the inflectional type elastic curve has a turning angle which is given by a formula,
involving the Jacobi sn function, of the form
sin
θ0
K
θ
= sin sn( (L − s), k)
2
2
L
16
and Sogo shows that
sin
θj
θ0
K
= sin sn( (N − j), k)
2
2
N
is the turning angle of an approximating discrete elastic curve.
Figure 16 shows (one of) the differentiable elastic splines with minimal
bending energy and length nine.
2.5
2.0
1.5
1.0
0.5
-0.5
0.5
1.0
1.5
2.0
2.5
3.0
Figure 16: Centered Splining
6
Comments
• The discretization of smooth curves and the splining of discrete curves
in three space is also well studied. Similar, though at times more involved, case by case constructions can be carried out in all six cases
discussed in detail for the curves in the plane. A complete understanding of circles and N-gons guides the discretization and splining methods
in the plane. Similarly by first carrying out the most basic case of the
helix one is able to succeed with curves in three space as well.
• The three settings (Inscribed, Circumscribed, and Centered) and only
these three settings, are used extensively in the literature. This is
true in both pure and applied differential geometry. Other settings we
considered, although formally feasible, are not as natural. For example
the reader can consider circles whose enclosed areas agree with the
enclosed areas of a regular N -gons.
17
• We feel there is no absolute “right” definition of discrete curvature
or torsion. A particular application may inform the researcher as to
which definitions to use. For example clothoids arise in the building
of highway off ramps. So in that case the circumscribed setting might
be more natural. In a more abstract context the circumscribed setting
is also used by T. Hoffman in his dissertation on discrete curves and
surfaces [7]. It seems clear that Gauss would have used the centered
setting, as it agrees most closely with his definition of the curvature
of an angle between two intersecting curves and with his definition of
curvature given by the normal Gauss map. This setting is used, for
example, by Doliwa and Santini [6] in their work on the integrable
dynamics of discrete curves.
• There is also a vast literature on discrete surface theory which goes back
over one hundred years. See [4] and references there. Not surprisingly
there is an even wider variety of definitions for the standard concepts
such as discrete Gauss curvature, discrete mean curvature, discrete
umbilics, etc. Again it seems clear that there is no absolute “right”
definition. How one chooses to define “the discretization” of a smooth
surface will again depend on which properties one wishes to preserve.
The theory of “integrable discretizations” in particular has been applied
to soap bubbles, minimal surfaces, Hasimoto surfaces (i.e. the surfaces
swept out by smoke-rings) and surfaces of constant Gauss curvature.
Similar comments apply to the theory of splining discrete surfaces.
• We have highlighted the Frenet frame because it is the most well known
curve framing. Discrete versions of the Bishop frame [2],[5] can also be
derived using the ideas of this paper. The Bishop frame is particularly
useful for curves that have points of zero curvature.
• We have considered only the simplest discretizations and the simplest
splinings. One which are as local as possible, taking into account only
the “nearest neighbors.” We feel the diversity and elegance of the cases
covered give a nice survey. Third order versions, either taking into
account more points for each calculation or by including curvature into
boundary conditions, can be found in the literature.
18
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19